[FOM] Godel Sentence

[Harvey Friedman]

Reply to Kanovei. 

On 8/23/03 12:13 AM, "Kanovei" <kanovei at wmwap1.math.uni-wuppertal.de> wrote:

> 
> As I had a chance to note in another occasion,
> the problem with "Goedel sentences" is not that they
> depend on the choice of the coding system in the background,
> but rather that the common interpretation of great Goedel
> results is wrong.
> In particular, the common belief that
> *there exist true, but unprovable sentences of PA*
> (allegedly by Goedel's theorems)
> is plain wrong in its straightforward sense, because the
> 25-centuries of development of mathematics under more or less
> current standards of rigor fail to produce anything near
> a mathematical sentence accepted to be TRUE
> (as a mathematical sentence)
> but NOT because its mathematical proof IS GIVEN
> (or at least, as for facts like 2+3=5, it is a common agreement
> that a proof can be maintained by anyone with a minimal
> knowledge of the subject).
> 
> Those who believe  in the paradigm
> *there exist true, but unprovable sentences of PA*
> in its straightforward sense
> are welcome to kindly present such a remarkable sentence
> along with a demonstration of its desired properties.
> 
> V.Kanovei
> _______________________________________________
>
Kanovei confuses two issues. There is the well known ROBUSTLY FORMULATED
theorem, call it T, of mathematics, fully accepted by the mathematics
community, that there exist true sentences in the language of PA, not
provable in PA. But then there is the issue of presenting an example that
meets certain criteria. In particular, there is the open question of whether
or not there is a reasonably short such sentence in an official primitive
language of PA. 

1. If we modify the definition of PA to allow abbreviation power, then one
can actually give completely readily understandable and universally
recognizable examples. One must bear in mind that with systems based on
abbreviation power, the notion of a statement has to be appropriately
modified. A statement is a well formed sentence together with the
abbreviation statements that lead up to that sentence.

2. One can give such an example of 1, using abbreviation power, that is
reasonably recognizable, directly thru Godel's work, however it is not very
satisfying mathematically.

3. More satisfying mathematically, and more directly recognizable, are the
various mathematical examples known, starting with Paris/Harrington, and
continuing with various other diverse examples which are yet more
mathematically satisfying. I plan to write a survey of these, as I have been
intensely involved in constructing new examples at the PA level for many
years. To make any of these mathematically recognizable, one must use PA
equipped with abbreviation power. Nevertheless, one should bear in mind the
well established theorem T.

4. Here is an interesting point I hadn't quite realized before. Let us say
that we want only to construct a SCHEME in either the primitive language of
PA, or in PA with abbreviations, which represents a true assertion that is
not provable in PA. We can take this in several good senses. The strongest
natural sense is that the scheme is true in the obvious second order sense,
but it has a substitution instance via formulas of PA that results in a
sentence that is not provable in PA. Then it would appear that the
complexity of the known examples is greatly reduced. I.e., the simplest
schemes in the primitive language of PA look to be much simpler than the
simplest sentences in the primitive language of PA. Also the simplest
schemes in PA with abbreviation power look to be much simpler than the
simplest sentences in PA with abbreviation power.

PS: I will be fully back on the FOM by October.

Harvey Friedman